Thirteenth-Century Chinese Mathematical Texts and Its Pedagogical Implications in the 21st Century: Suanxue qimeng and Yang Hui suanfa

Wann-Sheng Horng （洪萬生） Department of Mathematics National Taiwan Normal University, TAIWAN Suanxue qimeng and Yang Hui suanfa in East Asia

• Suanxue qimeng (by Zhu Shijie) and Yang Hui suanfa (by Yang Hui): the two most important texts of mathematics which greatly influenced how tongsan (東算 Korean mathematics) and wasan (和 算 Japanese mathematics) took their shape. • Big issue to which historians of mathematics in East Asia would pay a lot of attention despite that both traditions had already well developed on their own. • Reference: 兒玉明人（Kodama Akito），《十五世紀朝鮮刊 銅 活字版數學書》 HPM and Ancient Texts of Mathematics

• Taking an HPM perspective, these texts can also be introduced into mathematics class, either in college liberal study program or high school, in a very interesting way. • A few problems from these two texts demonstrating how HPM perspectives can shed new light to exploring their meanings (both mathematical and historical). Yanghui suanfa (楊輝算法)

• Yang Hui, also known as Qianguang, a native of Qiangtan (present-day Hangzhou) in Southern Song dynasty (1127-1280). He served local government as a low official responsible for accounting. • His collected work on mathematics, Yanghui suanfa, includes Chengchu tongbian suanbao (乘除通變本末), Fasuan chuyong bunmo, Xugu zhaichi suanfa (續古摘奇算法) and Tianmu bilei chengchu jiefa (田畝比類乘除捷法)

• A syllabus for mathematics learning (習算綱目) from Fasuan chuyong bunmo (法算取用本末, p. 54) • Area Formulas for Planar shape from Tianmu bilei chengchu jiefa (pp. 87- 89, 91) • Information about Korean edition (朝鮮板刻資料)(p. 97) Problem Solving in terms of the method of tianyuan in SQ

Zhu Shijie and His Suanxue qimeng (SQ)

• Zhu Shijie (朱世傑) of thirteenth-century Yuan dynasty, whose Siyuan yujian (四元玉鑑, Precious Mirror of the Four Elements, 1303) is regarded as representing the zenith of Chinese mathematics, came from some place near present-day Beijing. Yet nothing significant to his life can be traced back. • Suanxue qimeng was printed in 1299 and disappeared thereafter in China for more than five hundred years. In 1839 Qing Chinese scholar Luo Shilin (羅士琳) published an annotated version based on the Korean edition. And in what follows we can see the scan of a few pages of both Korean and Chinese editions.

• Suanxue qimeng (SQ)

• The Suanxue qimeng (Introduction to Mathematical Study) has three chapters, divided into 20 sections and 259 problems. • Starting from computation methods for multiplication and division and goes on to root extraction and the method of tian yuan (天元術, celestial element) – that is, solving polynomial equations). It includes almost all of the various aspects and content of mathematics as a branch of science of that time. • This book, a complete work going from the simple to the difficult, is indeed a very good textbook as an “introduction” (qimeng, 啟蒙). This may in part explain why Korean people used it, among others, to train their Chungin mathematicians (中人算學者). Suanxue qimeng (SQ)

• Contents and multiplication table (目錄與釋九數法)（p. 102） • Pi for ancient method (古法圓率)（p. 105） • Various 2-D diagrams and their area formulas such as trapezoid (梯 田), isosceles triangle (圭田), triangle (三斜田), and octagon (方五 斜七八角田)（pp. 124-125）

Suanxue qimeng (SQ)

• Chicken and rabbit in the same cage (雞兔同籠,Kikaku in Japanese) （p. 130） • Truncated pyramid with square base (方亭臺)、truncated cone (圓 亭臺): volume formula（p. 140） • Gaussian elimination method for solving system of linear equations (方程正負門)（p. 153） • Unlocking the method for solving equations (開方釋鎖門)（pp. 157, 159）

Problem Solving in terms of the method of tianyuan in SQ

Graphical Illustration: Li Huang (李潢, ?-1812), Jiuzhang suanshu xicao tushuo (《九章算術細草圖說》) Dissecting Cube: Kangxi emperor ed., Shuli jingyun (Collected Basic Principles of Mathematics, 1723) (康熙主編《數理精蘊》 正立方體切割) Yanghui suanfa vs. Suanxue qimeng

• Both texts contain no 3-D diagrams. • Following Liu Hui, Yang Hui uses 2-D diagram and out-in principle (以盈補虛或出入相補) as a tool to demonstrate area formula in plane while Zhu Shijie does not show any intention to do similar things. • In his section of unlocking the method for solving equations, Zhu Shijie presents a systematic exposition of the method of tianyuan. The method of setting up equations (setting up the tianyuan yi, 立 天元一) at that time was some sort of “high tech” which apparently was destined to exert its due influence over Korean and Japanese mathematicians. • In his Xiangjie jiuzhang suanfa zuanlei (詳解九章算法), Yang Hui reorganizes the contents of the nine chapters into a coherent system by designing a new standard of classification, indicating the significance of the method rather than the problem per se, which never be tried again by any Chinese mathematicians. And since it was not introduced to Korea and Japan, the text did not have a chance to make influence over the two neighbors. • Due to their influence, the nine chapters in China and their versions in Korea, Japan (?) and Vietnam came to form a textual community (九章文本社群), a term from Scott Montgomery who regards the role played by say Ptolemy’s Almagest in the Middle Age Europe. Contents of Jiuzhang suanshu (The Nine Chapters)《九章算術》

• Chapter One Fangtian (方田): simplifying fractional numbers (a Chinese version of Euclidean algorithm), arithmetical operations of fractional numbers, area formulas. • Sumi (粟米)：比例計算，例如今有術，已知 a : b = c : x，求x • Cuifen (衰分)：比例分配 • Shaoguang (少廣)：開平方，開立方法，球體積公式 • Shanggong (商功)：體積公式，例如陽馬術. • Jungshu (均輸)：複比例問題

• Yingbuzu (盈不足)：一次方程的算術解法，例如盈不足術 • Fangcheng (方程)：多元一次方程聯立消元解法，例如方程術，正負術 • Gougu (勾股)：勾股定理及其測量運用 Classification of the Nine Chapters

• The first six chapters were edited according to the characteristic of problems such as those of area measurement in chapter one while in the last three chapters the unknown editors put problems which are solved by the same method such as the method of fangcheng (Chapter 8) into the same chapter. 前六章按問題性質分類，後三 章按數學方法（所謂『術』）分類。 Yang Hui’s New Classification of the “Nine Chapters”

• 楊輝：「…殊不知〔九章〕所傳之本，一部的其真矣。如粟米章之互換，少 廣章之求由開方，皆重疊無謂，而作者題問不歸章次亦有之。今作纂類互見 目錄，以辨其訛，後之明者更為詳釋，不亦善乎。」 • 乘除第一：共40問，方田38+粟米2 • 互換第二：共55問，粟米31+衰分11+均輸11+盈朒（不足）2 • 合率第三 • 分率第四 • 衰分第五 • 壘積第六 • 盈不足第七 • 方程第八 • 句股第九 HPM: Examples

• Analogy (比類) in Yang Hui’s mathematics: Reasoning analogously! • An echo to Liu Hui’s reasoning (this) “class” by (that similar) class (以類推類) ? • Chicken and rabbit in the same cage (kikaku 雞兔同籠): synthesis (算術) vs. analysis (代數) and why Viete’s classic in algebraic symbolism entitled as “analytic art”? • How HPM and HM (History of Mathematics) can be beneficial to each other.

Hermeneutics

• How teachers implement these endeavors into their classes? • I would recommend, as Masami Isoda suggests for the theme of the math education session, an approach underlying which hermeneutics plays a key role. • Hermeneutic circles: Jahnke’s version • Hermeneutic tetrahedron H

M

I L O

Diagram 1. Jahnke’s Hermeneutic Twofold Circles. H: historian of mathematics; I: historical interpretation; M: ancient mathematician; L: mathematical theories; O: mathematical objects Diagram 2: A hermeneutic twofold circle for teacher’s teaching by means of textbook. T: teacher; I: teacher’s interpretation of textbook’s contents; E: editors of the textbook; S/K: curriculum standard and related mathematical knowledge; C: content knowledge. Diagram 3. Integrating history of mathematics into hermeneutic twofold circle. T: mathematics teacher; I: interpreting textbook contents; M: ancient mathematicians; L: mathematical theories; O: mathematical objects. Diagram 4. Hermeneutic Tetrahedron. 푪ퟏ: Primary circle for textbook editors, curriculum standard / mathematical theories and content knowledge. 푪ퟐ: Primary circle of Jahnke’s hermeneutic model. Diagram 5. Hermeneutic Tetrahedron Model Diagram 6. Cross Section of the Hermeneutic Tetrahedron. HPM: Examples

• Analogy (比類) in Yang Hui’s mathematics: Reasoning analogously! • Chicken and rabbit in the same cage (kikaku 雞兔同籠): synthesis (算術) vs. analysis (代數) and why Viete’s classic in algebraic symbolism entitled as “analytic art”? • How HPM and HM (History of Mathematics) can be beneficial to each other. A Story of Yu

• How and Why a (senior high school) teacher can accommodate himself/herself to be both teacher and historian? Hermeneutic approach and Pedagogical Reflection

• Teachers would firstly engage the students to read carefully the texts in the history of mathematics, then helped them to understand the meaning of the contents, and finally sought to make sense of the texts in terms of their historical contexts. • Analogy/contrast between teaching situations and historical context. • The teachers are encouraged to consult not only the students’ cognitive issues but also the historical “stranger’s” view of doing mathematics which is often difficult to make sense even with historian’s expertise. Mingde Team from New Taipei City

• Yu-Fen Chen and her team at the Mingde Senior High School at New Taipei City. • Yu-Fen is one of my former graduate students and her master thesis is an HPM study of junior high school geometry curriculum by a reference to Euclid’s Elements. References

• Dauben, Joseph, Xu Yibao and Guo Shuchun (2013). Nine Chapters on the Art of Mathematics. Shenyang: Liaoing Education Press. • Horng, Wann-Sheng (2004). “Teacher’s Professional Development in terms of the HPM: A Story of Yu”, presented to 2004 HPM, Uppsala, Sweden. • Isoda, Masami (2002). “Hermeneutics for Humanizing Mathematics Education”, Tsukuba Journal of Educational Studies in Mathematics 21. • Jahnke, Hans Niels (1994) “The Historical Dimension of Mathematical Understanding: Objectifying the Subjective”, Proceedings of the 18th International Conference for the Psychology of Mathematics Education vol. I, Lisbon: University of Lisbon, pp. 139-156. • Jahnke, Hans Niels et al., (2000). “The Use of Original Sources in the Mathematics Classroom”, in Chapter 9 of John Fauvel and Jan van Maanen (eds.), History in Mathematics Education, Dordrecht / Boston / London: Kluwer Academic Publishers, pp. 291-728. • Li Yan and Du Shiran (1987). Chinese Mathematics: A Concise History. Oxford: Clarendon Press. • Martzloff, Jean-Claude (1997). A History of Chinese Mathematics. Berlin/Heidelberg/New York: Springer-Verlag.